Geodesic rays in the uniform infinite half-planar quadrangulation return to the boundary

TL;DR

This paper proves that all geodesic rays in the UIHPQ intersect the boundary infinitely often, with sparse intersection points, and shows similar behavior in the UIH triangulation, advancing understanding of geodesic structures in random planar maps.

Contribution

It demonstrates that geodesic rays in the UIHPQ intersect the boundary infinitely often and are proper, also establishing similar properties in the UIH triangulation.

Findings

Geodesic rays in UIHPQ intersect the boundary infinitely many times.

Intersection points are sparsely distributed along the boundary.

Geodesic rays in UIH triangulation behave similarly to those in UIHPQ.

Abstract

We show that all geodesic rays in the uniform infinite half-planar quadrangulation (UIHPQ) intersect the boundary infinitely many times, answering thereby a recent question of Curien. However, the possible intersection points are sparsely distributed along the boundary. As an intermediate step, we show that geodesic rays in the UIHPQ are proper, a fact that was recently established by Caraceni and Curien (2015) by a reasoning different from ours. Finally, we argue that geodesic rays in the uniform infinite half-planar triangulation behave in a very similar manner, even in a strong quantitative sense.